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XVI. IMPERFECT COMPETITION
This inequality can be satisfied for a value of $\delta$, $0 < \delta < 1$, provided $\delta < .5$. Put differently,
if $\delta > .5$, then (2) is not satisfied and firm 1 will not find its interests served by unilaterally
departing from cooperative play in any period. Thus, we see that tit-for-tat is an equilibrium
strategy for the firms provided the discount factor is sufficiently close to 1.
7. Problems and Complements
The problems are grouped by topic starting with one-shot markets.
7.1. One-Shot Market Games
The first group of problems asks you to find equilibria in one-shot Cournot and Bertrand
Duopoly games.
• Problem 7.1 Consider a duopoly market where the inverse market demand is specified
by $Q = 36 - 2P$. The marginal cost of production is assumed to be zero for both firms.
Calculate the Cournot equilibrium for a Cournot Duopoly game.
• Problem 7.2 Derive a $2 \times 2$ payoff matrix for the Cournot Duopoly game using the results
of the first problem. Find the game's solution using the arrow diagram.
